28 SCIENTIFIC APPARATUS. 



Indian ritualists were led to the problem " To find two square 

 numbers of which the sum shall also be a square " by the 

 existence of a religious feeling which required that altars of differ- 

 ent shapes should have the same superficial area. By these and 

 similar inquiries, they were brought into contact with many 

 questions of mensuration, and learned to solve them by approxi- 

 mate methods of considerable exactness ; the value, for example, 

 which they obtained for the side of a square equal to a circle of 

 given diameter is correct as far as the third decimal place inclu- 

 sively. Contemporary records of these researches still exist, and 

 though they tell us of a time when science was in its infancy, 

 they bear emphatic testimony to the genius and patient industry 

 of the ancient workers. They are further characterized by that 

 predominance of the arithmetical above the geometrical spirit, 

 which forms so marked a contrast between the mathematical 

 tendencies of India and of Greece. But while in these earlier 

 treatises we can watch the growth of mathematical conceptions, 

 called forth and fostered by the practical requirements of the old 

 Vedic ceremonial, the purely scientific study of geometry and 

 arithmetic in India belongs to a later period, probably to the 

 fourth century after our era. Even then, the Hindus were the first 

 to discover the method of solving indeterminate equations of 

 the first degree, a method which was not known in Europe till 

 the seventeenth century, and perhaps not demonstrated till the 

 eighteenth. But the crowning achievement of Indian mathe- 

 matical genius was the solution of the problem known as the 

 Pellian Equation, upon which the analysis of indeterminate 

 equations of the second degree may be said entirely to depend. 

 The Indian mathematicians gave no demonstration of their 

 solution. That demonstration was first given, at least fourteen 

 hundred years later, by Lagrange, one of the greatest of European 

 mathematicians, and the memoir in which he has recorded this 

 discovery has always been regarded as one of the principal monu- 

 ments of his genius. The indeterminate equation of the first 



